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AP Statistics

Learn about confidence intervals and how to calculate them for population means. Explore the conditions required for constructing confidence intervals and interpret the results. Examples and calculations provided.

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AP Statistics

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  1. AP Statistics Chapter 10 Notes

  2. Confidence Interval • Statistical Inference: Methods for drawing conclusions about a population based on sample data. • Level C Confidence Interval (2 parts) • 1. Confidence interval calculated from the data. • Estimate ± margin of error • 2. Confidence level – gives the probability that the interval will capture the true parameter value in repeated samples. (most often 95%)

  3. Conditions for constructing a CI (for μ) • Randomness: Data should come from an SRS. • Not always realistic. As long as our sampling methods are reasonable, we will have to assume that the data can be treated as an SRS. • Independence: N > 10n • Sampling distribution of is approx Normal

  4. Critical Values • Values (z*) that mark off a specified area under the Normal curve.

  5. Confidence Interval for a Population Mean • Choose an SRS of size n from a population having an unknown mean μ and known standard deviation σ. A level C confidence interval for μ is…

  6. Steps for Constructing a CI • 1. Identify the population and parameter of interest. • 2. Verify that all conditions are met. Name the type of interval. • 3. Do confidence interval calculations. • 4. Interpret the results in context.

  7. Example • Suppose that the standard deviation of heart rate for all 18yr old males is 10 bpm. A random sample of 50 18-year-old males yields a mean of 72 beats per minute. • (a) Construct and interpret a 95% confidence interval for the mean heart rate μ. • (b) Construct and interpret a 90% CI. • (c) Construct and interpret a 99% CI.

  8. Interpretation • We are 95% confident that the true mean heart rate of all 18 year old males is between 69.23 bpm and 74.77 bpm. • What does 95% confidence mean? • 95% of the samples taken from the population will yield an interval which contains the true population mean heart rate.

  9. Margin of Error • Margin of Error gets smaller when… • z* gets smaller. (lower z* = less confident) • σ gets smaller. (not easy to do in reality) • n gets larger. • Using the heart rate example, what would my sample size need to be if I want a 95% confidence interval with a margin of error, m, of only 1 beat per minute?

  10. Interval for unknown σ • If we don’t know σ, (we usually don’t), we can estimate σ by using s, the sample standard deviation. • is called the standard error of the sample mean . • Known σ z distribution (Standard Normal) • Never changes • Unknown σ t distribution (t(k)) • Changes based on its degrees of freedom k = n - 1

  11. One Sample t-interval • A level C confidence interval for μ is • t* is the critical value for the t(n – 1) distribution.

  12. Paired t Procedures • Used to compare the responses to the two treatments in a matched pairs design or to the before and after measurements on the same subjects. • The parameter μd in a paired t procedure is the mean difference in response. • Robust: accurate even when conditions are not met. • t procedures are not robust against outliers but are robust against Non-Normality.

  13. Confidence Interval for p • Conditions: • SRS • Independence: N > 10n • and are > 10. • Confidence interval for unknown p.

  14. Finding sample size • To find the sample size needed for a desired C and m… • p* is a guessed value for p-hat. If you have no educated guess, then say p* = .5.

  15. Reminders • The margin of error only accounts for random sampling error. Non-response, undercoverage, and response bias must still be considered. • Random sampling: allows us to generalize the results to a larger population. • Random assignment: allows us to investigate treatment effects.

  16. Confidence Interval Summary • 1. State the population and the parameter. • 2. Explain how each condition is/isn’t met. • (a) SRS • (b) Independence: N > 10n. • (c) Normality: • For p: and are > 10. • For μ: Look for large n. (Central Limit Theorem) • If n is small, look to see if the data were sampled from a Normal population. At last resort, look at the sample data to make sure that there are no outliers or strong skewness.

  17. Summary Continued • 3. Calculate the confidence interval. • Estimate ± margin of error

  18. Summary Continued • 4. Interpret the interval in context. • We are ____% confident that the true population mean/proportion of ____________ falls between ( , ). • If you are asked to interpret the confidence level… • ______% of the samples taken from the population yield an interval which contains the true population mean/proportion.

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